Transport Phenomena in Biological Systems A Textbook for

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					            Transport Phenomena in Biological Systems: A Textbook for
                            Biomedical Engineers

                      George A. Truskey, Fan Yuan and David F. Katz
                          Department of Biomedical Engineering
                                     Duke University
                                   Durham, NC 27708

    Understanding the physical, chemical and biological processes governing the movement of mass and
transmission of forces throughout an organism is important in biomedical engineering and physiology.
Transport processes influence the normal and pathological function of cells and organs. For the
biomedical engineer, transport processes are important in the design and operation of a number of
devices including those for kidney dialysis, removal of toxins, hybrid artificial organs, and drug delivery.
Thus, this discipline is an important component of the undergraduate curriculum in biomedical

    At Duke, we find that the traditional transport texts are not particularly helpful. Although these texts
are often quite good for their intended audience in chemical or mechanical engineering, engineering
concepts are not presented in the context of important biological problems. The absence of a sufficient
presentation of biological concepts and problems limits the relevance of these texts and causes difficulties
for students in applying the principles in all but the simplest problems. Based upon our experience, we
have developed a set of course material consisting of chapters from selected texts, notes, and articles
from the current literature. Although this assembly of material is an improvement upon the texts
available, this approach is limited because students do not receive enough examples and the literature
papers often do not present enough material for students to easily comprehend the application of
transport models. Based upon these concerns we have decided to develop a textbook for a two semester
undergraduate course sequence on transport phenomena in biological systems.

     Our goals for the text on transport phenomena in biological systems are to present engineering
fundamentals and biological applications in a unified way and to provide necessary skills to develop and
critically analyze models of biological processes. The book will cover topics in fluid mechanics, mass
transport and biochemical interactions (chemical reactions and binding interactions). The engineering
concepts presented are motivated by specific biological problems. Applications of these concepts are
discussed immediately after development of the engineering concepts. In this way the student will gain
understanding of the specific topic presented as well as the application of the concept to important
biological problems. Each chapter will contain a significant number of example and homework problems
which elaborate upon problems discussed in the text or address new biomedical problems. Problems and
examples will include analytical as well as numerical solutions. We will emphasize analytical solutions
because these solutions can often provide important physical insights important for introductory
material, even if the insight provides only a first order level of understanding. We will also cite current
literature for those interested in more detailed analysis of problems of current interest.

    We assume that the text will be used by students in their junior or senior year. We assume some
exposure to biology but do provide some review material and reference relevant texts in cell biology and
physiology. The students have not had any previous exposure to mass and momentum transport except
for a brief discussion of diffusion in introductory chemistry. Although students will have been exposed
to most of the mathematical concepts discussed in the text, we provide a review in the Appendix. We
have found the review material very helpful in the course we have taught.

Text Outline The text contains an introduction and three parts. The Introduction provides the
motivation for the text and its organization. The three parts examine concepts and biomedical
applications in fluid mechanics, mass transport, and the effect of transport processes upon biochemical
interactions. Each part begins with a chapter developing the relevant engineering concepts in terms of a
specific biological context. In subsequent chapters, more detailed examples are considered so that
students can understand the process of taking a biological problem and developing a model to address
quantitative questions. Although the text is designed for a two semester course, the text could be adapted
for a single semester course by omitting some chapters which present detailed applications (e.g. chapters
4, 8-10, 15 and 16).

    Biological problems examined include those at the molecular level (e.g. protein diffusion), cellular
level (e.g. transport across endothelium), tissue level (e.g. transport in tumors or across the glomerulus),
organ level (e.g. blood flow in organs) and organism level (e.g. pharmacokinetic models). Many of the
problems are at the cellular and tissue level and fewer focus upon the organ and organism level. The
reasons for this distribution are because understanding many problems at the cellular and tissue level
simplifies development of organ level models. The problems themselves will involve analytical and
numerical solutions of transport problems.      Numerical solutions are provided with MATLAB although
the problems will be general enough for solution by any number of packages. The outline is as follows.

    Chapter 1. Introduction
                1.1.    Rationale and motivation
                1.2.    Definition of transport processes
                1.3.    Relative importance of convection and diffusion
                1.4.    Transport across and within cells
                                 1.4.1. Transport across the cell membrane
                                 1.4.2. Transport within the cell
                1.5.    Transcellular transport
                                 1.5.1. Epithelial cells
                                 1.5.2. Endothelial cells
                1.6.    Physiological transport systems
                        1.6.1    The cardiovascular system
                        1.6.2    Respiratory system
                        1.6.3    Gastrointestinal system
                        1.6.4    Liver
                        1.6.5    Kidneys
                        1.6.6    Integrated organ function
                1.7.    Application of Transport Processes in Disease Pathology, Treatment and Device
                        1.7.1    Transport processes and atherosclerosis
                        1.7.2    Transport processes and cancer treatment
                        1.7.3    Transport processes and tissue engineering
                1.8.    Relative importance of transport and reaction processes
                1.9.    Overview of text
                1.10.   Questions
                1.11.   Problems
                1.12.   References

Part A. Fundamentals and Applications of Fluid Mechanics
              2.1.    Introduction
              2.2.    Kinematics
                      2.2.1. Control Volumes
                      2.2.2. Velocity Field
                      2.2.3. Flow rate
                      2.2.4. Acceleration
                      2.2.5. Streamlines, Streamtubes and Streaklines
              2.3.    Conservation Relations
                      2.3.1. Conservation of Mass
                      2.3.2. Momentum Balances
                      2.3.3. Forces
                      2.3.4. Boundary Conditions
              2.4.    Fluid Statics
                      2.4.1. Static Equilibrium
                      2.4.2. Surface Tension
                      2.4.3. Membrane and Cortical Tension
                2.5.     Constitutive Relations
                         2.5.1. Newton's Law of Viscosity
                         2.5.2. Non-Newtonian Rheology
                         2.5.3. Time Dependent Viscoelastic Behavior
                2.6.     Laminar and Turbulent flow
                2.7.     Application of Momentum Balances
                         2.7.1. Flow induced by a sliding plate
                         2.7.2. Pressure-driven flow through a narrow rectangular channel
                         2.7.3. Flow of a fluid through a cylindrical tube
                         2.7.4. Flow of a power law fluid in a cylindrical tube
                         2.7.5. Flow Between Rotating cylinders
                2.8.     Differential Form of the Conservation of Mass in Three Dimensions
                2.9.     Differential Form of the Conservation of Linear Momentum and Navier-
                         Stokes Equation in Three Dimensions
                2.10.    Rheology and Flow of Blood
                         2.10.1 Measurement of Blood Viscosity
                         2.10.2. Rheology of Blood Flow in Large Tubes
                         2.10.3. Fahreus-Lindquist effect
                2.11.    Questions
                2.12.    Problems
                2.13.    References

    Chapter 3. Dimensional Analysis and Scaling
                3.1. Dimensional analysis and dimensionless groups
                3.2. Scaling and development of models to study flow in arteries
                3.3. Low Reynolds number flow
                3.4. Cell swimming
                3.5. Flow in porous media
                3.6. Fluid transport in the vitreous
                3.7. Lubrication theory and flow of red cells in capillaries

    Chapter 4. Fluid Flow in the Circulation and Tissues
                4.1. Steady and unsteady flow in arteries
                4.2. Flow in branched and curved vessels
                4.3. Arterial fluid dynamics and atherosclerosis
                4.4. Flow in the venous circulation
                4.5. Flow in capillary networks
                4.6. Flow in porous tissues

    Chapter 5. Turbulent Flow and Macroscopic Balances for Momentum Transport
                5.1. Turbulence
                5.2. Turbulence in physiological fluid mechanics
                5.3. Integral forms of the conservation of mass and linear momentum
                5.4. Bernoulli's equation
                5.5. Flow and pressure drop across heart valves

B. Fundamentals and Applications of Mass Transport
    Chapter 6. Introduction to Mass Transport
                6.1. Definitions and Fluxes
                6.2. Diffusion as a Random Walk
                6.3. Constitutive Relations
                         6.3.1. Fick's Law of Diffusion for Dilute Solutions
                         6.3.2. Multicomponent Diffusion; Stefan-Maxwell equation
                6.4. Estimation of Diffusion Coefficients in Solution
                         6.4.1. Stokes-Einstein Relation and Prediction of Diffusion Coefficients
                         6.4.2. Correlations
                6.5. Steady State Diffusion in One-Dimension
                         6.5.1. Diffusion in Rectangular Coordinates
                         6.5.2. Diffusion in Cylindrical Coordinates
                         6.5.3. Diffusion in Spherical Coordinates
                6.6. Unsteady Diffusion
                         6.6.1. One-Dimensional Diffusion in a Semi-Infinite Medium
                         6.6.2. One-Dimensional Unsteady Diffusion in a Finite Media
                         6.6.3. Diffusion of a Solute into a Sphere from a Well-Stirred Bath
                6.7. Quasi-steady transport across membranes and determination of membrane
                6.8. Diffusion-Limited Reactions
                         6.8.1. Diffusion-Limited Binding and Dissociation in Solution
                         6.8.2. Diffusion-Limited Binding between a Cell Surface Protein and a Solute
                         6.8.3. Diffusion-limited Binding on a Cell Surface

    Chapter 7. Conservation of Mass for Dilute Solutions
               7.1. Generalized conservation relations for binary systems
               7.2. Dimensional analysis
               7.3. Electrolyte transport and ion transport across membranes
               7.4. Diffusion and convection
               7.5. Interphase mass transport and mass transfer coefficients
               7.6. Boundary layer theory and correlations for mass transfer coefficients

    Chapter 8. Transport in Porous Media
                8.1. Factors influencing solvent and solute transport in tissue
                8.2. Models of transport in porous media

    Chapter 9. Fluid and solute transport across endothelium and in tissues
                9.1. Endothelial structure and summary of experimental observations
                9.2. Models of solute transport across capillaries
                9.3. Models of macromolecular transport in tissues

    Chapter 10. Solvent and solute transport across the kidney glomerulus
                10.1. Anatomy and physiology of the kidney and glomerulus
                10.2. Role of osmotic pressure in solvent flow across the kidney
                10.3. Role of epithelial cells and basement membrane in transport across the glomerulus

    Chapter 11. Macroscopic Balances for Mass Transport
                11.1. Integral forms of the conservation of mass and linear momentum
                11.2. Bernoulli's equation
                11.3. Flow and pressure drop across heart valves
                11.4. Integral forms of the conservation of mass for dilute solution
                11.5. Mass transport and design of an artificial kidney

Part C. The Effect of Mass Transport upon Biochemical Interactions
    Chapter 12. Mass Transport and Biochemical Interactions
                12.1. Chemical kinetics and reaction mechanisms
                12.2. Enzyme kinetics and antigen-antibody interactions
                12.3. Heterogeneous chemical reactions
                12.4. Combined mass transport and chemical reactions
                12.5. Effect of flow upon local concentrations in blood
                12.6. Lipoprotein transport and atherosclerosis

    Chapter 13. Oxygen transport from the lungs to tissues
                13.1. Oxygen-hemoglobin binding kinetics and equilibrium
                13.2. Dynamics of oxygen transport and binding in the capillaries
                13.3. Oxygen delivery to tissues
                13.4. Heterogeneity of oxygen transport

    Chapter 14. Ligand-receptor kinetics on the cell surface and molecular transport within cells
            14.1. Diffusion on cell membranes and through cells
            14.2. Diffusion and binding on cell surfaces
            14.3. Kinetic models of receptor-mediated endocytosis and receptor-ligand trafficking
            14.4. Oxygen transport through cells and reaction in mitochondria

Chapter 15. Cell adhesion and cell signaling
           15.1. Kinetic and thermodynamic models of cell adhesion
           15.2. Effect of diffusion on cell adhesion
           15.3. Cell signaling models
           15.4. Role of diffusion in G protein activation

Chapter 16. Transport of drugs and macromolecules in tumors
            17.1. Tumor structure
            17.2. Transport properties
            17.3. Models of transport processes
            17.4. Effect of binding and drug metabolism upon drug uptake by tumors

Chapter 17. Transport in Organs and Organisms
            18.1. Flow and transport based models
            18.2. Effect of heterogeneous capillary distribution upon transport in organs
            18.3. Scaling laws and relation to transport properties
            18.4. Physiologically based pharmacokinetic models
            18.5 Pharmacokinetic models and drug delivery

Appendix. Relevant Mathematical Concepts
          A.1. Review of calculus and solution of ordinary differential equations
          A.2. Solution of partial differential equations
                  A.2.a. Characteristic value problems
                  A.2.b. Separation of variables
                  A.2.c. Laplace Transform methods
          A.3. Basics of vectors and tensors
          A.4. Numerical solution of ordinary and partial differential equations using MATLAB